Biological enzyme catalysis is the name for life's ability to rapidly invoke coordinated rearrangements of molecular structures using highly specialised, relatively-large functional proteins called enzymes. It is well established that enzymes reduce the activation energy barrier, which vastly accelerates the rate of reaction, however the specific dynamics and quantum mechanics are largely uncharted. Recent experimentation and modelling hints that there is a phase transition dynamic that is fundamental to enzymatic catalysis. The hypothesis that I explore in this thesis is that a mathematical model of a substrate molecule based upon anharmonic quantum vibrational modes (molecular phonons) will predict phase transition dynamics due to the presence of the appropriate enzyme. A molecular phonon is one particular individualised wave pattern in the locations of atomic nuclei from the total molecular dance that spans the low-frequency bulk swaying motions, mid-frequency bond stretches, to the high-frequency oscillations of the hydrogen bond network. In this thesis, I derive a generalised set of equations of motion for the anharmonic response behaviour of molecular phonons to enzyme stimulus via a dissipative quantum master equation, and I explore the phase transition dynamics by determining the stability bifurcation of the steady states. I then determine the critical minimum stimulus required for a molecular vibrational mode to exhibit a phase transition. The primary catalytic reaction of focus for this thesis is the cleavage of the glycosidic carbon-oxygen bond of the complex sugar isomaltose into two individual glucose molecules via the action of the enzyme MalL. Prior to this thesis, the atomic structure of MalL was obtained experimentally via X-ray diffraction crystallography of prepared MalL crystals, then digitised into the Gromacs software package and energy minimised in a simulated water bath to approximate the vibrational modes of the in vivo configuration. I obtain the substrate molecule's harmonic frequencies and anharmonic shifts via molecular dynamics performed using the Gaussian 09 normal mode analysis and simulated anharmonic infrared spectroscopy packages, respectively. I construct a generalised coding package in Matlab to extract the molecule and enzyme data, perform a quantum anharmonic analysis of the molecule--enzyme complex and their interactions, and then simulate the response of the molecule to stimulus from the enzyme. I determine that the vibrational modes of isomaltose that significantly contribute to stretching the glycosidic carbon-oxygen bond occur within the 21 to 35 THz (700 to 1,170 cm^{-1}) frequency bandwidth, and the dominant stretching modes are contained within 28 to 33 THz (930 to 1,100cm^{-1}), which is in agreement with the experimental literature of polysaccharides. In the final chapter, I perform numerical simulations of the equations of motion to compute the energy response spectrogram for a proxy substrate (benzene) under conditions that are informed by literature and the theoretical findings of this thesis. In lieu of anharmonic isomaltose data, I selected benzene as it is a small molecule with a high degree of symmetry which has several bond stretching modes within the aforementioned bandwidth. The results show that while the stimulus from water is insufficient, there are vibrational modes pertaining to glycosidic bond stretching that can be selectively excited to phase transition behaviour due to a reasonable stimulus from the supply of phonons from the MalL enzyme.